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In an A.P if ${a_2} = 13;{a_4} = 3$ find ${a_1},{a_3}$

Last updated date: 13th Jun 2024
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Hint: An arithmetic progression or AP is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.
This fixed term is called the common difference of the AP.
Here we need to find the terms by using the $n^{th}$ term formula.

Formula used: Formula for $nth$ term, ${a_n} = {a_1} + (n - 1)d$
Here, ${a_n} = $The $n^{th}$ term in the sequence
${a_1} = $ The first term in the sequence
$d = $ The common difference of the sequence

Complete step-by-step answer:
It is given that in the AP its terms are ${a_2} = 13,{a_4} = 3$
We have to find the value of ${a_1},{a_3} = ?$
As we know that the formula for $nth$ term
Now we are put the values of given for finding the values for required variables
Using the $nth$ term formula
${a_2} = {a_1} + (n - 1)d = a + d$
Once again using the $nth$ term formula
${a_4} = {a_1} + (4 - 1)d = a + 3d$
Here ${a_1} + d = 13.........(1)$
${a_1} + 3d = 3.......(2)$
On subtracting the equations $(2) - (1)$ and we get,
$\left( {{a_1} + 3d = 3} \right) - \left( {{a_1} + d = 13} \right)$
Cancelling the same terms and subtracts it we get,
$ \Rightarrow 2d = - 10$
Now we divide the terms
$ \Rightarrow $$d = \dfrac{{ - 10}}{2}$
Hence, $d = - 5$
Now we need to put the value of $d$ in the equation $(1)$
${a_1} - 5 = 13$
We take the numbers in the same side that is right hand side
${a_1} = 13 + 5$
After adding the both the numbers
Thus, we get the required term
$\therefore {a_1} = 18$
Therefore, ${a_1} = 18$
Now we applying the formula and put the values of given for finding the values for required variables
${a_3} = {a_1} + 2d$
$ = 18 + 2( - 5)$
On multiplying the bracket terms and we get,
$ = 18 - 10$
On subtracting we get,
$\therefore {a_3} = 8$

Hence, the value of ${a_1} = 18$ and ${a_3} = 8$

Note: A sequence is said to be A.P if and only if the common difference between the consecutive terms remains constant throughout the series.
It is always advisable to remember all the series related formula whether it is of $n^{th}$ term or of sum of $n^{th}$ as it helps saving a lot of time.
Remember that the common difference of AP can be positive, negative or zero.